Archive for the ‘Mathematics’ Category

In this post, we discuss the interesting recent paper by Steve Ross,”The Recovery Theorem”, in which a method is proposed to disentangle the risk aversion component from the subjective probability measure from state prices. In particular, a method is proposed to back out the market’s forecast of returns (a distribution over returns) from option prices. Attached is the pdf summary detailing the results.

We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.

Definition 2.1A function on an open set is meromorphic if there exists a discrete set of points such that is holomorphic on and has poles at each . Furthermore, is meromorphic in the extended complex plane if is either meromorphic or holomorphic at . In this case we say that has a pole or is holomorphic at infinity.

By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.

Proposition 2.2Let be the discrete set of singularities of a complex function where is an open set in . For a fixed , suppose the Laurent expansion for in an annulus about is given by . Then,

The function has a removable singularity at if and only if for all .

The function has a pole at if and only if there exists with such that for all ; that is, the Laurent expansion of about has only finitely many negative terms.

The function has an essential singularity at if and only if the Laurent expansion of about has infinitely many negative terms.

Furthermore, is meromorphic on the extended complex plane if and only if there exists such that for .

The following series of posts comprises our introduction to complex analysis as taught by Professor Rowan Killip at the University of California, Los Angeles, during the Fall quarter of 2009. Where necessary, course notes have been supplemented with details written by the authors of this website using assistance from Complex Analysis by Elias Stein and Rami Shakarchi. The basic properties of complex numbers will be assumed allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

The basic properties of complex numbers will be assumed, allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

Definition 1.1Suppose is an open set and . We say is holomorphic (or complex-differentiable) at if there exists We say is holomorphic on if it has this property for all .

We can rewrite this formula in terms of the real and imaginary parts of to surmise the relationship between complex differentiability and real analytic differentiability. Let with and , and write where . Then,

We first notice that this is stronger than the differentiability of the real map in . In the real, multivariable case, the derivative of this map is a linear operator, namely, the Jacobian, ; in our equation above, the matrix on the right hand side is . Clearly, it is endowed with a distinct structure summarized in the following proposition.